演講者：蕭錫源 A new BEM formulation for transient axisymmetric poroelasticity via particular integrals K.H. Park a, P.K. Banerjee b,*

Abstract A simple particular integral formulation is presented for the first time in a purely axisymmetric poroelastic analysis. A simple particular integral formulation is presented for the first time in a purely axisymmetric poroelastic analysis. The axisymmetric elastostatic and steady-state potential flow equations are used as the complementary solution. The axisymmetric elastostatic and steady-state potential flow equations are used as the complementary solution. The particular integrals for displacement, traction, pore pressure and flux are derived by integrating three-dimensional formulation alongthe circumferential direction leading to elliptic integrals. The particular integrals for displacement, traction, pore pressure and flux are derived by integrating three-dimensional formulation alongthe circumferential direction leading to elliptic integrals.

1. Introduction The general theory of poroelasticity is governed by two coupled di ff erential equations: the Navier equation with pore pressure body force and the pore fluid flow equation as (Banerjee, 1994) The general theory of poroelasticity is governed by two coupled di ff erential equations: the Navier equation with pore pressure body force and the pore fluid flow equation as (Banerjee, 1994)

是位移 是有效滲透率 是孔隙壓力 和 Lame’s 的常數 the undrained 不透水的 該排水體積彈性模量 是在對堅實組成部分的 BULK 模數某些情況經驗得知的常數 是 body force 和 source （如果存在的話） for 2D (3D)

常數 和 也可以表示在不排水體積的 彈性模數 (Rice and Cleary, 1976) 是著名的 skempton 係數的孔隙壓力。 是著名的 skempton 係數的孔隙壓力。

Park and Banerjee (2002a) first proposed the particular integral formulation

左為２００６ 下為２００２

2.Three-dimensional particular integral formulation 位移 曳引力 孔隙壓力 流量

將右式 global shape function 代入

將上三頁的式子代入２００６的 非均值式子中得下列係數

3. Axisymmetric particular integral formulation For axisymmetric problems, use of such polynomial functions as functions of r and z coordinates have been discussed in Henry et al. (1987). For axisymmetric problems, use of such polynomial functions as functions of r and z coordinates have been discussed in Henry et al. (1987). It is of considerable interest to note that Wang and Banerjee (1988, 1990) in their developments of particular integrals in free-vibration analysis of axisymmetric solids also observed the same to be true. It is of considerable interest to note that Wang and Banerjee (1988, 1990) in their developments of particular integrals in free-vibration analysis of axisymmetric solids also observed the same to be true.

為方便起見，所界定的軸對稱 case

考慮到在圓柱坐標系的純粹的軸對稱體

then 分別為在 X 點在 R 和 Z - 方向 分別為在 X 點在 R 和 Z - 方向 的 normal vector 。 的 normal vector 。

4. Numerical implementation 軸對稱彈性力學和穩定狀態勢流方程的根本解 分別代表 jump terms resulting 所產生的奇異性質的 和 和

離散 ↓ 離散 ↓ ↓

因考慮而加入 有限的數量、時間、位移、牽引、 孔隙壓力和流量

5. Numerical examples Example 1

Example 2

其中 是 的根

Example 3

6. Conclusions The simple particular integral formulation has been developed for axisymmetric coupled poroelastic analysis. The simple particular integral formulation has been developed for axisymmetric coupled poroelastic analysis. The equations of axisymmetric elastostatic and steady-state potential flow have been used as the complementary functions. The equations of axisymmetric elastostatic and steady-state potential flow have been used as the complementary functions. The particular integrals of displacement, traction, pore pressure and flux are obtained by integrating three-dimensional BEM formulation along the circumferential direction and converting them into elliptic integrals. The particular integrals of displacement, traction, pore pressure and flux are obtained by integrating three-dimensional BEM formulation along the circumferential direction and converting them into elliptic integrals.